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ISU Logic Seminar: Closed groups generated by generic measure preserving transformations

Author: Lona

Speaker: Slawomir Solecki (Cornell)


The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation T T has emerged. This picture included substantial evidence that pointed to these groups (for a generic T T ) being all topologically isomorphic to a single group, namely, L0L^0—the topological group of all Lebesgue measurable functions from [0,1][0,1] to the circle. In fact, Glasner and Weiss asked if this is the case.

We will describe the background touched on above (including all the relevant definitions) and outline a proof of the following theorem that answers the Glasner–Weiss question in the negative: for a generic measure preserving transformation TT, the closed group generated by TT is not topologically isomorphic to L0 L^0 . The proof rests on an analysis of unitary representations of L0 L^0 .